Challenge Mode is a special mode that unlocks upon completing the game for the first time and is accessible from the main menu. It is essentially New Game+ and the majority of your things will carry over into it, with the exception of a few gadgets that are received during the story. The enemies in Challenge Mode are tougher than they were originally, dealing some more damage and even having more health themselves. The catch about Challenge Mode is that two things will unlock, both of which will greatly benefit you.
Upon reaching level 5 with any weapon, it will change into a new form, usually granting it an additional property that wasn't on it before. With a weapon at level 5 and having the Holocard set for that weapon type, you will be able to purchase the Omega variant in Challenge Mode. Not only will this allow you to level a weapon to 10, but it also unlocks new nodes on the upgrade screen, so you will be able to spend your excess Raritanium.
The other thing that unlocks with Challenge Mode is the Bolt Multiplier, and it functions exactly as it sounds. As you kill enemies in your adventure, you will notice a multiplier appear next to your bolts in the upper right corner of the screen. The maximum multiplier is x20, but there is a slight catch to it. Should you ever get hit by an enemy, then the multiplier will reset back to zero and you'll have to start all over again.